Construct Non-Hierarchical P/NBD Model for Short Timeframe Synthetic Data

In this workbook we construct the non-hierarchical P/NBD models on the CD-NOW transaction data.

1 Load and Construct Datasets

1.1 Load Short-Timeframe Synthetic Transaction Data

We now want to load the CD-NOW transaction data.

Code

customer_cohortdata_tbl <- read_rds("data/shortsynth_customer_cohort_data_tbl.rds")
customer_cohortdata_tbl |> glimpse()

Rows: 5,000
Columns: 5
$ customer_id     <chr> "SFC202001_0001", "SFC202001_0002", "SFC202001_0003", …
$ cohort_qtr      <chr> "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1", "2020 Q1",…
$ cohort_ym       <chr> "2020 01", "2020 01", "2020 01", "2020 01", "2020 01",…
$ first_tnx_date  <dttm> 2020-01-01 02:15:58, 2020-01-01 00:40:17, 2020-01-01 …
$ total_tnx_count <int> 3, 9, 1, 2, 5, 1, 4, 6, 2, 4, 1, 11, 3, 3, 1, 1, 11, 3…

Code

customer_transactions_tbl <- read_rds("data/shortsynth_transaction_data_tbl.rds")
customer_transactions_tbl |> glimpse()

Rows: 29,454
Columns: 7
$ customer_id   <fct> SFC202001_0002, SFC202001_0001, SFC202001_0004, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 00:40:17, 2020-01-01 02:15:58, 2020-01-01 18…
$ tnx_dow       <fct> Wed, Wed, Wed, Wed, Thu, Thu, Thu, Thu, Thu, Thu, Thu, F…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0001", "T20200101-0002", "T20200101-0003", "T…
$ tnx_amount    <dbl> 51.73, 139.05, 11.72, 88.30, 1.16, 0.95, 127.85, 35.07, …

Code

customer_subset_id <- read_rds("data/shortsynth_customer_subset_ids.rds")
customer_subset_id |> glimpse()

 Factor w/ 5000 levels "SFC202001_0002",..: 2 3 8 10 14 16 17 21 25 27 ...

We re-produce the visualisation of the transaction times we used in previous workbooks.

Code

plot_tbl <- customer_transactions_tbl |>
  group_nest(customer_id, .key = "cust_data") |>
  filter(map_int(cust_data, nrow) > 3) |>
  slice_sample(n = 30) |>
  unnest(cust_data)

ggplot(plot_tbl, aes(x = tnx_timestamp, y = customer_id)) +
  geom_line() +
  geom_point() +
  labs(
      x = "Date",
      y = "Customer ID",
      title = "Visualisation of Customer Transaction Times"
    ) +
  theme(axis.text.y = element_text(size = 10))

1.2 Load Derived Data

Code

obs_fitdata_tbl   <- read_rds("data/shortsynth_obs_fitdata_tbl.rds")
obs_validdata_tbl <- read_rds("data/shortsynth_obs_validdata_tbl.rds")

customer_fit_stats_tbl <- obs_fitdata_tbl |>
  rename(x = tnx_count)

1.3 Load Subset Data

We also want to construct our data subsets for the purposes of speeding up our valuations.

Code

customer_fit_subset_tbl <- obs_fitdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_fit_subset_tbl |> glimpse()

Rows: 1,000
Columns: 6
$ customer_id    <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC2020…
$ first_tnx_date <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 1…
$ last_tnx_date  <dttm> 2020-02-20 14:43:57, 2020-01-10 17:49:12, 2020-07-17 0…
$ tnx_count      <dbl> 2, 1, 5, 4, 0, 10, 10, 0, 0, 0, 6, 2, 0, 0, 0, 1, 0, 3,…
$ t_x            <dbl> 7.2170614, 1.2841927, 28.0650995, 12.3070761, 0.0000000…
$ T_cal          <dbl> 104.4151, 104.3210, 104.1918, 104.1751, 104.0693, 104.0…

Code

customer_valid_subset_tbl <- obs_validdata_tbl |>
  filter(customer_id %in% customer_subset_id)

customer_valid_subset_tbl |> glimpse()

Rows: 1,000
Columns: 3
$ customer_id       <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC2…
$ tnx_count         <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ tnx_last_interval <dbl> NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, NA, …

We now use these datasets to set the start and end dates for our various validation methods.

Code

dates_lst <- read_rds("data/shortsynth_simulation_dates.rds")

use_fit_start_date <- dates_lst$use_fit_start_date
use_fit_end_date   <- dates_lst$use_fit_end_date

use_valid_start_date <- dates_lst$use_valid_start_date
use_valid_end_date   <- dates_lst$use_valid_end_date

We now split out the transaction data into fit and validation datasets.

Code

customer_fit_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_fit_start_date,
    tnx_timestamp <= use_fit_end_date
    )
  
customer_fit_transactions_tbl |> glimpse()

Rows: 5,016
Columns: 7
$ customer_id   <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 15…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 139.05, 11.72, 35.07, 54.66, 70.30, 91.17, 393.76, 588.8…

Code

customer_valid_transactions_tbl <- customer_transactions_tbl |>
  filter(
    customer_id %in% customer_subset_id,
    tnx_timestamp >= use_valid_start_date,
    tnx_timestamp <= use_valid_end_date
    )
  
customer_valid_transactions_tbl |> glimpse()

Rows: 1,509
Columns: 7
$ customer_id   <fct> SFC202108_0025, SFC202008_0057, SFC202112_0120, SFC20211…
$ tnx_timestamp <dttm> 2022-01-01 00:01:40, 2022-01-01 01:06:39, 2022-01-01 05…
$ tnx_dow       <fct> Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sat, Sun, Sun, Sun, S…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20220101-0001", "T20220101-0002", "T20220101-0007", "T…
$ tnx_amount    <dbl> 2.97, 69.73, 0.36, 4.88, 725.98, 22.79, 45.67, 0.60, 8.1…

Finally, we want to extract the first transaction for each customer, so we can add this data to assess our models.

Code

customer_initial_tnx_tbl <- customer_fit_transactions_tbl |>
  slice_min(n = 1, order_by = tnx_timestamp, by = customer_id)

customer_initial_tnx_tbl |> glimpse()

Rows: 1,000
Columns: 7
$ customer_id   <fct> SFC202001_0001, SFC202001_0004, SFC202001_0008, SFC20200…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 18:04:32, 2020-01-02 15…
$ tnx_dow       <fct> Wed, Wed, Thu, Thu, Fri, Fri, Fri, Sat, Sat, Sun, Sun, M…
$ tnx_month     <fct> Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, Jan, J…
$ tnx_week      <chr> "00", "00", "00", "00", "00", "00", "00", "00", "00", "0…
$ invoice_id    <chr> "T20200101-0002", "T20200101-0003", "T20200102-0004", "T…
$ tnx_amount    <dbl> 139.05, 11.72, 35.07, 54.66, 70.30, 91.17, 393.76, 588.8…

We now expand out these initial transactions so that we can append them to our simulations.

Code

sim_init_tbl <- customer_initial_tnx_tbl |>
  transmute(
    customer_id,
    draw_id       = list(1:n_sim),
    tnx_timestamp,
    tnx_amount
    ) |>
  unnest(draw_id)

sim_init_tbl |> glimpse()

Rows: 2,000,000
Columns: 4
$ customer_id   <fct> SFC202001_0001, SFC202001_0001, SFC202001_0001, SFC20200…
$ draw_id       <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 1…
$ tnx_timestamp <dttm> 2020-01-01 02:15:58, 2020-01-01 02:15:58, 2020-01-01 02…
$ tnx_amount    <dbl> 139.05, 139.05, 139.05, 139.05, 139.05, 139.05, 139.05, …

Before we start on that, we set a few parameters for the workbook to organise our Stan code.

Code

stan_modeldir <- "stan_models"
stan_codedir  <-   "stan_code"

2 Fit First P/NBD Model

We now construct our Stan model and prepare to fit it with our synthetic dataset.

We also want to set a number of overall parameters for this workbook

To start the fit data, we want to use the 1,000 customers. We also need to calculate the summary statistics for the validation period.

2.1 Compile and Fit Stan Model

We now compile this model using CmdStanR.

Code

pnbd_fixed_stanmodel <- cmdstan_model(
  "stan_code/pnbd_fixed.stan",
  include_paths =   stan_codedir,
  pedantic      =           TRUE,
  dir           =  stan_modeldir
  )

We then use this compiled model with our data to produce a fit of the data.

Code

stan_modelname <- "pnbd_short_fixed1"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")

stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 1.00,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed1_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )
  
  pnbd_short_fixed1_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed1_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed1_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -51501.25 -51500.60 63.94 63.90 -51609.92 -51399.30 1.01      675
 lambda[1]      0.44      0.42  0.16  0.15      0.21      0.73 1.00     2471
 lambda[2]      0.21      0.18  0.13  0.11      0.05      0.46 1.00     2548
 lambda[3]      0.27      0.21  0.22  0.17      0.04      0.72 1.00     1820
 lambda[4]      0.13      0.08  0.16  0.09      0.00      0.46 1.00     1604
 lambda[5]      0.49      0.43  0.26  0.24      0.16      0.98 1.00     2233
 lambda[6]      0.25      0.20  0.20  0.16      0.03      0.64 1.00     1989
 lambda[7]      0.14      0.08  0.19  0.09      0.00      0.50 1.00     2173
 lambda[8]      0.16      0.15  0.07  0.06      0.07      0.29 1.00     2698
 lambda[9]      0.39      0.35  0.22  0.19      0.12      0.81 1.00     2264
 ess_tail
      993
     1195
      984
      834
      954
     1367
     1047
      943
     1585
     1390

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed1_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed1-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

2.2 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed1_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We also check \(N_{eff}\) as a quick diagnostic of the fit.

Code

pnbd_short_fixed1_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed1_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

2.3 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed1_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed1_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed1",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5010
  )

pnbd_short_fixed1_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed1_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed1_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed1_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed1_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed1_assess_valid_simstats_tbl.rds"

2.3.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed1_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

2.3.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed1_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

3 Fit Alternate Prior Model.

We want to try an alternate prior model with a smaller co-efficient of variation to see what impact it has on our procedures.

Code

stan_modelname <- "pnbd_short_fixed2"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 0.50,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed2_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed2_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed2_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed2_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -109786.10 -109785.00 59.51 60.79 -109884.05 -109690.00 1.00
 lambda[1]       0.37       0.36  0.11  0.10       0.22       0.56 1.00
 lambda[2]       0.23       0.21  0.10  0.09       0.10       0.40 1.00
 lambda[3]       0.25       0.23  0.12  0.11       0.10       0.46 1.00
 lambda[4]       0.21       0.19  0.11  0.10       0.07       0.41 1.00
 lambda[5]       0.34       0.32  0.13  0.12       0.16       0.59 1.00
 lambda[6]       0.24       0.23  0.12  0.11       0.09       0.47 1.01
 lambda[7]       0.21       0.19  0.11  0.10       0.07       0.41 1.00
 lambda[8]       0.19       0.18  0.06  0.07       0.10       0.30 1.00
 lambda[9]       0.31       0.30  0.12  0.12       0.15       0.53 1.00
 ess_bulk ess_tail
      602      977
     4075     1038
     4211     1301
     4550     1199
     4672     1266
     4808     1415
     4067     1229
     3742     1343
     4541     1533
     3819     1507

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed2_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed2-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

3.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed2_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed2_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed2_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

3.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed2_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed2_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed2",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5020
  )

pnbd_short_fixed2_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed2_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed2_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed2_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed2_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed2_assess_valid_simstats_tbl.rds"

3.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed2_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

3.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed2_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

4 Fit Tight-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Code

stan_modelname <- "pnbd_short_fixed3"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.10,
    mu_cv     = 0.50,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed3_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed3_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed3_stanfit <- read_rds(stanfit_object_file)
}
  
pnbd_short_fixed3_stanfit$print()

  variable      mean    median    sd   mad        q5       q95 rhat ess_bulk
 lp__      -84963.91 -84963.90 61.63 62.57 -85062.73 -84863.49 1.00      586
 lambda[1]      0.45      0.43  0.15  0.14      0.24      0.72 1.00     3162
 lambda[2]      0.21      0.18  0.13  0.12      0.05      0.45 1.00     2741
 lambda[3]      0.26      0.20  0.21  0.16      0.04      0.67 1.00     3077
 lambda[4]      0.13      0.08  0.16  0.09      0.01      0.45 1.00     1996
 lambda[5]      0.49      0.44  0.26  0.24      0.15      0.97 1.00     2726
 lambda[6]      0.24      0.19  0.19  0.15      0.03      0.61 1.00     2271
 lambda[7]      0.14      0.09  0.17  0.10      0.01      0.47 1.00     2556
 lambda[8]      0.16      0.16  0.07  0.06      0.07      0.28 1.00     2656
 lambda[9]      0.40      0.36  0.21  0.19      0.13      0.80 1.01     3067
 ess_tail
      872
     1002
     1361
     1194
     1145
     1266
      891
     1245
     1163
     1250

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed3_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed3-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

4.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed3_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed3_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed3_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

4.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed3_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed3_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed3",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5030
  )

pnbd_short_fixed3_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed3_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed3_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed3_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed3_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed3_assess_valid_simstats_tbl.rds"

4.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed3_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

4.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed3_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

5 Fit Narrow-Short-Lifetime Model

We now want to try a model where we use priors with a tighter coefficient of variation for lifetime but keep the CoV for transaction frequency.

Code

stan_modelname <- "pnbd_short_fixed4"
stanfit_seed   <- stanfit_seed + 1
stanfit_prefix <- str_c("fit_", stan_modelname) 

stanfit_object_file <- glue("data/{stanfit_prefix}_stanfit.rds")


stan_data_lst <- customer_fit_stats_tbl |>
  select(customer_id, x, t_x, T_cal) |>
  compose_data(
    lambda_mn = 0.25,
    lambda_cv = 1.00,
    
    mu_mn     = 0.20,
    mu_cv     = 0.30,
    )

if(!file_exists(stanfit_object_file)) {
  pnbd_short_fixed4_stanfit <- pnbd_fixed_stanmodel$sample(
    data            =                stan_data_lst,
    chains          =                            4,
    iter_warmup     =                          500,
    iter_sampling   =                          500,
    seed            =                 stanfit_seed,
    save_warmup     =                         TRUE,
    output_dir      =                stan_modeldir,
    output_basename =               stanfit_prefix,
    )

  pnbd_short_fixed4_stanfit$save_object(stanfit_object_file, compress = "gzip")

} else {
  message(glue("Found file {stanfit_object_file}. Loading..."))
  
  pnbd_short_fixed4_stanfit <- read_rds(stanfit_object_file)
}

pnbd_short_fixed4_stanfit$print()

  variable       mean     median    sd   mad         q5        q95 rhat
 lp__      -139178.73 -139179.00 59.46 58.56 -139280.00 -139084.00 1.00
 lambda[1]       0.45       0.43  0.16  0.15       0.23       0.74 1.00
 lambda[2]       0.22       0.19  0.15  0.12       0.05       0.50 1.00
 lambda[3]       0.28       0.22  0.21  0.17       0.05       0.67 1.00
 lambda[4]       0.15       0.10  0.16  0.10       0.01       0.45 1.00
 lambda[5]       0.50       0.45  0.27  0.25       0.15       1.01 1.00
 lambda[6]       0.26       0.22  0.20  0.17       0.04       0.67 1.00
 lambda[7]       0.16       0.10  0.18  0.11       0.01       0.53 1.00
 lambda[8]       0.17       0.16  0.07  0.07       0.07       0.30 1.01
 lambda[9]       0.42       0.38  0.23  0.21       0.12       0.86 1.00
 ess_bulk ess_tail
      768     1163
     3432     1406
     2935      876
     3045     1397
     2365      975
     3307     1281
     2378     1114
     2490      870
     2918     1026
     2779     1249

 # showing 10 of 9961 rows (change via 'max_rows' argument or 'cmdstanr_max_rows' option)

We have some basic HMC-based validity statistics we can check.

Code

pnbd_short_fixed4_stanfit$cmdstan_diagnose()

Processing csv files: /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-1.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-2.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-3.csvWarning: non-fatal error reading adaptation data
, /home/rstudio/btydwork/stan_models/fit_pnbd_short_fixed4-4.csvWarning: non-fatal error reading adaptation data


Checking sampler transitions treedepth.
Treedepth satisfactory for all transitions.

Checking sampler transitions for divergences.
No divergent transitions found.

Checking E-BFMI - sampler transitions HMC potential energy.
E-BFMI satisfactory.

Effective sample size satisfactory.

Split R-hat values satisfactory all parameters.

Processing complete, no problems detected.

5.1 Visual Diagnostics of the Sample Validity

Now that we have a sample from the posterior distribution we need to create a few different visualisations of the diagnostics.

Code

parameter_subset <- c(
  "lambda[1]", "lambda[2]", "lambda[3]", "lambda[4]",
  "mu[1]",     "mu[2]",     "mu[3]",     "mu[4]"
  )

pnbd_short_fixed4_stanfit$draws(inc_warmup = FALSE) |>
  mcmc_trace(pars = parameter_subset) +
  expand_limits(y = 0) +
  labs(
    x = "Iteration",
    y = "Value",
    title = "Traceplot of Sample of Lambda and Mu Values"
    ) +
  theme(axis.text.x = element_text(size = 10))

We want to check the \(N_{eff}\) statistics also.

Code

pnbd_short_fixed4_stanfit |>
  neff_ratio(pars = c("lambda", "mu")) |>
  mcmc_neff() +
    ggtitle("Plot of Parameter Effective Sample Sizes")

Finally, we want to check out the energy diagnostic, which is often indicative of problems with the posterior mixing.

Code

pnbd_short_fixed4_stanfit |>
  nuts_params() |>
  mcmc_nuts_energy(binwidth = 50)

5.2 Assess the Model

As we intend to run the same logic to assess each of our models, we have combined all this logic into a single function run_model_assessment, to run the simulations and combine the datasets.

We first run the assessment data.

Code

pnbd_stanfit <- pnbd_short_fixed4_stanfit |>
  recover_types(customer_fit_stats_tbl)

pnbd_short_fixed4_assess_data_lst <- run_model_assessment(
  model_stanfit       = pnbd_stanfit,
  insample_tbl        = customer_fit_subset_tbl,
  fit_label           = "pnbd_short_fixed4",
  fit_end_dttm        = use_fit_end_date     |> as.POSIXct(),
  valid_start_dttm    = use_valid_start_date |> as.POSIXct(),
  valid_end_dttm      = use_valid_end_date   |> as.POSIXct(),
  precompute_rootdir  = "precompute",
  data_dir            = "data",
  summary_include_tnx = FALSE,
  sim_seed            = 5040
  )

pnbd_short_fixed4_assess_data_lst |> glimpse()

List of 5
 $ model_fit_index_filepath     : 'glue' chr "data/pnbd_short_fixed4_assess_fit_index_tbl.rds"
 $ model_valid_index_filepath   : 'glue' chr "data/pnbd_short_fixed4_assess_valid_index_tbl.rds"
 $ model_simstats_filepath      : 'glue' chr "data/pnbd_short_fixed4_assess_model_simstats_tbl.rds"
 $ model_fit_simstats_filepath  : 'glue' chr "data/pnbd_short_fixed4_assess_fit_simstats_tbl.rds"
 $ model_valid_simstats_filepath: 'glue' chr "data/pnbd_short_fixed4_assess_valid_simstats_tbl.rds"

5.2.1 Check In-Sample Data Validation

We first check the model against the in-sample data.

Code

simdata_tbl <- pnbd_short_fixed4_assess_data_lst |>
  use_series(model_fit_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  bind_rows(sim_init_tbl) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_fit_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

This fit looks reasonable and appears to capture most of the aspects of the data used to fit it. Given that this is a synthetic dataset, this is not surprising, but at least we appreciate that our model is valid.

5.2.2 Check Out-of-Sample Data Validation

We now repeat for the out-of-sample data.

Code

simdata_tbl <- pnbd_short_fixed4_assess_data_lst |>
  use_series(model_valid_index_filepath) |>
  read_rds() |>
  use_series(sim_file) |>
  map_dfr(read_rds) |>
  select(customer_id, draw_id, sim_data) |>
  unnest(sim_data) |>
  arrange(customer_id, draw_id, tnx_timestamp)


assess_plots_lst <- create_model_assessment_plots(
  obsdata_tbl = customer_valid_transactions_tbl,
  simdata_tbl = simdata_tbl
  )

assess_plots_lst |> map(print)

$total_plot

$quant_plot

6 Compare Model Outputs

We have looked at each of the models individually, but it is also worth looking at each of the models as a group.

We now want to combine both the fit and valid transaction sets to calculate the summary statistics for both.

Code

obs_summstats_tbl <- list(
    fit   = customer_fit_transactions_tbl,
    valid = customer_valid_transactions_tbl
    ) |>
  bind_rows(.id = "assess_type") |>
  group_by(assess_type) |>
  calculate_transaction_summary_statistics() |>
  pivot_longer(
    cols      = !assess_type,
    names_to  = "label",
    values_to = "obs_value"
    )

obs_summstats_tbl |> glimpse()

Rows: 16
Columns: 3
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "v…
$ label       <chr> "p10", "p25", "p50", "p75", "p90", "p99", "total_count", "…
$ obs_value   <dbl> 1.00000, 1.00000, 2.00000, 5.00000, 12.00000, 45.01000, 50…

Code

model_assess_transactions_tbl <- dir_ls("data", regexp = "pnbd_short_fixed.*_assess_.*index") |>
  enframe(name = NULL, value = "file_path") |>
  mutate(
    model_label = str_replace(file_path, "data/pnbd_short_(.*?)_assess_.*", "\\1"),
    assess_type = if_else(str_detect(file_path, "_assess_fit_"), "fit", "valid"),
    
    assess_data = map(
      file_path, construct_model_assessment_data,
      
      .progress = "construct_assess_data"
      )
    ) |>
  select(model_label, assess_type, assess_data) |>
  unnest(assess_data)

model_assess_transactions_tbl |> glimpse()

Rows: 30,568,892
Columns: 6
$ model_label   <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed…
$ assess_type   <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", …
$ customer_id   <fct> SFC202001_0001, SFC202001_0001, SFC202001_0001, SFC20200…
$ draw_id       <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 4, 4, 4, 4, 4, 4, 4, 4,…
$ tnx_timestamp <dttm> 2020-02-27 04:24:25, 2020-03-23 07:41:18, 2020-07-02 04…
$ tnx_amount    <dbl> 83.09, 0.73, 79.75, 33.17, 23.00, 89.13, 198.72, 22.91, …

We now want to calculate the transaction statistics on this full dataset, for each separate draw.

Code

model_assess_tbl <- model_assess_transactions_tbl |>
  group_by(model_label, assess_type, draw_id) |>
  calculate_transaction_summary_statistics()

model_assess_tbl |> glimpse()

Rows: 16,000
Columns: 11
$ model_label <chr> "fixed1", "fixed1", "fixed1", "fixed1", "fixed1", "fixed1"…
$ assess_type <chr> "fit", "fit", "fit", "fit", "fit", "fit", "fit", "fit", "f…
$ draw_id     <int> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,…
$ p10         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p25         <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ p50         <dbl> 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3…
$ p75         <dbl> 8, 8, 7, 9, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 7, 7, 7, 7, 8, 8…
$ p90         <dbl> 14.4, 18.0, 16.6, 19.0, 15.0, 16.0, 17.0, 17.0, 18.0, 18.0…
$ p99         <dbl> 57.00, 57.40, 46.32, 53.00, 56.82, 49.06, 44.84, 48.00, 49…
$ total_count <int> 4195, 4242, 4358, 4474, 4320, 4006, 4090, 4146, 4161, 4441…
$ mean_count  <dbl> 6.911038, 7.177665, 6.862992, 7.444260, 7.081967, 6.698997…

We now combine all this data to create a number of different comparison plots for the various summary statistics.

Code

#! echo: TRUE

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "total_count", "Total Transactions"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "mean_count", "Average Transactions per Customer"
  )

Code

create_multiple_model_assessment_plot(
  obs_summstats_tbl, model_assess_tbl,
  "p99", "99th Percentile Count"
  )

6.1 Write Assessment Data to Disk

We now want to save the assessment data to disk.

Code

model_assess_tbl |> write_rds("data/assess_data_pnbd_short_fixed_tbl.rds")

R Environment

Code

options(width = 120L)
sessioninfo::session_info()

─ Session info ───────────────────────────────────────────────────────────────────────────────────────────────────────
 setting  value
 version  R version 4.3.1 (2023-06-16)
 os       Ubuntu 22.04.3 LTS
 system   x86_64, linux-gnu
 ui       X11
 language (EN)
 collate  en_US.UTF-8
 ctype    en_US.UTF-8
 tz       Europe/Dublin
 date     2023-11-20
 pandoc   3.1.1 @ /usr/local/bin/ (via rmarkdown)

─ Packages ───────────────────────────────────────────────────────────────────────────────────────────────────────────
 package        * version    date (UTC) lib source
 abind            1.4-5      2016-07-21 [1] RSPM (R 4.3.0)
 arrayhelpers     1.1-0      2020-02-04 [1] RSPM (R 4.3.0)
 backports        1.4.1      2021-12-13 [1] RSPM (R 4.3.0)
 base64enc        0.1-3      2015-07-28 [1] RSPM (R 4.3.0)
 bayesplot      * 1.10.0     2022-11-16 [1] RSPM (R 4.3.0)
 bit              4.0.5      2022-11-15 [1] RSPM (R 4.3.0)
 bit64            4.0.5      2020-08-30 [1] RSPM (R 4.3.0)
 bridgesampling   1.1-2      2021-04-16 [1] RSPM (R 4.3.0)
 brms           * 2.20.4     2023-09-25 [1] RSPM (R 4.3.0)
 Brobdingnag      1.2-9      2022-10-19 [1] RSPM (R 4.3.0)
 cachem           1.0.8      2023-05-01 [1] RSPM (R 4.3.0)
 callr            3.7.3      2022-11-02 [1] RSPM (R 4.3.0)
 checkmate        2.3.0      2023-10-25 [1] RSPM (R 4.3.0)
 cli              3.6.1      2023-03-23 [1] RSPM (R 4.3.0)
 cmdstanr       * 0.6.0.9000 2023-11-07 [1] Github (stan-dev/cmdstanr@a13c798)
 coda             0.19-4     2020-09-30 [1] RSPM (R 4.3.0)
 codetools        0.2-19     2023-02-01 [2] CRAN (R 4.3.1)
 colorspace       2.1-0      2023-01-23 [1] RSPM (R 4.3.0)
 colourpicker     1.3.0      2023-08-21 [1] RSPM (R 4.3.0)
 conflicted     * 1.2.0      2023-02-01 [1] RSPM (R 4.3.0)
 cowplot        * 1.1.1      2020-12-30 [1] RSPM (R 4.3.0)
 crayon           1.5.2      2022-09-29 [1] RSPM (R 4.3.0)
 crosstalk        1.2.0      2021-11-04 [1] RSPM (R 4.3.0)
 curl             5.1.0      2023-10-02 [1] RSPM (R 4.3.0)
 digest           0.6.33     2023-07-07 [1] RSPM (R 4.3.0)
 directlabels   * 2023.8.25  2023-09-01 [1] RSPM (R 4.3.0)
 distributional   0.3.2      2023-03-22 [1] RSPM (R 4.3.0)
 dplyr          * 1.1.3      2023-09-03 [1] RSPM (R 4.3.0)
 DT               0.30       2023-10-05 [1] RSPM (R 4.3.0)
 dygraphs         1.1.1.6    2018-07-11 [1] RSPM (R 4.3.0)
 ellipsis         0.3.2      2021-04-29 [1] RSPM (R 4.3.0)
 evaluate         0.22       2023-09-29 [1] RSPM (R 4.3.0)
 fansi            1.0.5      2023-10-08 [1] RSPM (R 4.3.0)
 farver           2.1.1      2022-07-06 [1] RSPM (R 4.3.0)
 fastmap          1.1.1      2023-02-24 [1] RSPM (R 4.3.0)
 forcats        * 1.0.0      2023-01-29 [1] RSPM (R 4.3.0)
 fs             * 1.6.3      2023-07-20 [1] RSPM (R 4.3.0)
 furrr          * 0.3.1      2022-08-15 [1] RSPM (R 4.3.0)
 future         * 1.33.0     2023-07-01 [1] RSPM (R 4.3.0)
 generics         0.1.3      2022-07-05 [1] RSPM (R 4.3.0)
 ggdist           3.3.0      2023-05-13 [1] RSPM (R 4.3.0)
 ggplot2        * 3.4.4      2023-10-12 [1] RSPM (R 4.3.0)
 globals          0.16.2     2022-11-21 [1] RSPM (R 4.3.0)
 glue           * 1.6.2      2022-02-24 [1] RSPM (R 4.3.0)
 gridExtra        2.3        2017-09-09 [1] RSPM (R 4.3.0)
 gtable           0.3.4      2023-08-21 [1] RSPM (R 4.3.0)
 gtools           3.9.4      2022-11-27 [1] RSPM (R 4.3.0)
 hms              1.1.3      2023-03-21 [1] RSPM (R 4.3.0)
 htmltools        0.5.6.1    2023-10-06 [1] RSPM (R 4.3.0)
 htmlwidgets      1.6.2      2023-03-17 [1] RSPM (R 4.3.0)
 httpuv           1.6.12     2023-10-23 [1] RSPM (R 4.3.0)
 igraph           1.5.1      2023-08-10 [1] RSPM (R 4.3.0)
 inline           0.3.19     2021-05-31 [1] RSPM (R 4.3.0)
 jsonlite         1.8.7      2023-06-29 [1] RSPM (R 4.3.0)
 knitr            1.44       2023-09-11 [1] RSPM (R 4.3.0)
 labeling         0.4.3      2023-08-29 [1] RSPM (R 4.3.0)
 later            1.3.1      2023-05-02 [1] RSPM (R 4.3.0)
 lattice          0.21-8     2023-04-05 [2] CRAN (R 4.3.1)
 lifecycle        1.0.3      2022-10-07 [1] RSPM (R 4.3.0)
 listenv          0.9.0      2022-12-16 [1] RSPM (R 4.3.0)
 loo              2.6.0      2023-03-31 [1] RSPM (R 4.3.0)
 lubridate      * 1.9.3      2023-09-27 [1] RSPM (R 4.3.0)
 magrittr       * 2.0.3      2022-03-30 [1] RSPM (R 4.3.0)
 markdown         1.11       2023-10-19 [1] RSPM (R 4.3.0)
 Matrix           1.5-4.1    2023-05-18 [2] CRAN (R 4.3.1)
 matrixStats      1.0.0      2023-06-02 [1] RSPM (R 4.3.0)
 memoise          2.0.1      2021-11-26 [1] RSPM (R 4.3.0)
 mime             0.12       2021-09-28 [1] RSPM (R 4.3.0)
 miniUI           0.1.1.1    2018-05-18 [1] RSPM (R 4.3.0)
 munsell          0.5.0      2018-06-12 [1] RSPM (R 4.3.0)
 mvtnorm          1.2-3      2023-08-25 [1] RSPM (R 4.3.0)
 nlme             3.1-162    2023-01-31 [2] CRAN (R 4.3.1)
 parallelly       1.36.0     2023-05-26 [1] RSPM (R 4.3.0)
 pillar           1.9.0      2023-03-22 [1] RSPM (R 4.3.0)
 pkgbuild         1.4.2      2023-06-26 [1] RSPM (R 4.3.0)
 pkgconfig        2.0.3      2019-09-22 [1] RSPM (R 4.3.0)
 plyr             1.8.9      2023-10-02 [1] RSPM (R 4.3.0)
 posterior      * 1.4.1      2023-03-14 [1] RSPM (R 4.3.0)
 prettyunits      1.2.0      2023-09-24 [1] RSPM (R 4.3.0)
 processx         3.8.2      2023-06-30 [1] RSPM (R 4.3.0)
 promises         1.2.1      2023-08-10 [1] RSPM (R 4.3.0)
 ps               1.7.5      2023-04-18 [1] RSPM (R 4.3.0)
 purrr          * 1.0.2      2023-08-10 [1] RSPM (R 4.3.0)
 quadprog         1.5-8      2019-11-20 [1] RSPM (R 4.3.0)
 QuickJSR         1.0.7      2023-10-15 [1] RSPM (R 4.3.0)
 R6               2.5.1      2021-08-19 [1] RSPM (R 4.3.0)
 Rcpp           * 1.0.11     2023-07-06 [1] RSPM (R 4.3.0)
 RcppParallel     5.1.7      2023-02-27 [1] RSPM (R 4.3.0)
 readr          * 2.1.4      2023-02-10 [1] RSPM (R 4.3.0)
 reshape2         1.4.4      2020-04-09 [1] RSPM (R 4.3.0)
 rlang          * 1.1.1      2023-04-28 [1] RSPM (R 4.3.0)
 rmarkdown        2.25       2023-09-18 [1] RSPM (R 4.3.0)
 rstan            2.32.3     2023-10-15 [1] RSPM (R 4.3.0)
 rstantools       2.3.1.1    2023-07-18 [1] RSPM (R 4.3.0)
 rstudioapi       0.15.0     2023-07-07 [1] RSPM (R 4.3.0)
 rsyslog        * 1.0.3      2023-05-08 [1] RSPM (R 4.3.0)
 scales         * 1.2.1      2022-08-20 [1] RSPM (R 4.3.0)
 sessioninfo      1.2.2      2021-12-06 [1] RSPM (R 4.3.0)
 shiny            1.7.5.1    2023-10-14 [1] RSPM (R 4.3.0)
 shinyjs          2.1.0      2021-12-23 [1] RSPM (R 4.3.0)
 shinystan        2.6.0      2022-03-03 [1] RSPM (R 4.3.0)
 shinythemes      1.2.0      2021-01-25 [1] RSPM (R 4.3.0)
 StanHeaders      2.26.28    2023-09-07 [1] RSPM (R 4.3.0)
 stringi          1.7.12     2023-01-11 [1] RSPM (R 4.3.0)
 stringr        * 1.5.0      2022-12-02 [1] RSPM (R 4.3.0)
 svUnit           1.0.6      2021-04-19 [1] RSPM (R 4.3.0)
 tensorA          0.36.2     2020-11-19 [1] RSPM (R 4.3.0)
 threejs          0.3.3      2020-01-21 [1] RSPM (R 4.3.0)
 tibble         * 3.2.1      2023-03-20 [1] RSPM (R 4.3.0)
 tidybayes      * 3.0.6      2023-08-12 [1] RSPM (R 4.3.0)
 tidyr          * 1.3.0      2023-01-24 [1] RSPM (R 4.3.0)
 tidyselect       1.2.0      2022-10-10 [1] RSPM (R 4.3.0)
 tidyverse      * 2.0.0      2023-02-22 [1] RSPM (R 4.3.0)
 timechange       0.2.0      2023-01-11 [1] RSPM (R 4.3.0)
 tzdb             0.4.0      2023-05-12 [1] RSPM (R 4.3.0)
 utf8             1.2.4      2023-10-22 [1] RSPM (R 4.3.0)
 V8               4.4.0      2023-10-09 [1] RSPM (R 4.3.0)
 vctrs            0.6.4      2023-10-12 [1] RSPM (R 4.3.0)
 vroom            1.6.4      2023-10-02 [1] RSPM (R 4.3.0)
 withr            2.5.1      2023-09-26 [1] RSPM (R 4.3.0)
 xfun             0.40       2023-08-09 [1] RSPM (R 4.3.0)
 xtable           1.8-4      2019-04-21 [1] RSPM (R 4.3.0)
 xts              0.13.1     2023-04-16 [1] RSPM (R 4.3.0)
 yaml             2.3.7      2023-01-23 [1] RSPM (R 4.3.0)
 zoo              1.8-12     2023-04-13 [1] RSPM (R 4.3.0)

 [1] /usr/local/lib/R/site-library
 [2] /usr/local/lib/R/library

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Code

options(width = 80L)